For $0<\theta<\pi / 2$, if the eccentricity of the hyperbola $\mathrm{x}^2-\mathrm{y}^2 \operatorname{cosec}^2 \theta=5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^2 \operatorname{cosec}^2 \theta+y^2=5$, then the value of $\theta$ is :

The eccentricity of the conjugate hyperbola of the hyperbola ${x^2} - 3{y^2} = 1$, is

lf $e_1$ , $e_2$ and $e_3$ are eccentricities of the conics $y = {x^2} - x + 3,\,\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{3{a^4}}} = 1$ and ${a^2}{x^2} - 3{a^4}{y^2} = 1$ respectively, then which of the following is correct ? (where $a > 1)$

The locus of the point of instruction of the lines $\sqrt 3 x - y - 4 \sqrt 3 t = 0$  $\&$  $\sqrt 3tx + ty - 4\sqrt 3 = 0$  (where $ t$  is a parameter) is a hyperbola whose eccentricity is

The product of the lengths of perpendiculars drawn from any point on the hyperbola ${x^2} - 2{y^2} - 2 = 0$ to its asymptotes is

A point on the curve $\frac{{{x^2}}}{{{A^2}}} - \frac{{{y^2}}}{{{B^2}}} = 1$ is